14
Chemical
Kinetics
Visualizing Concepts
14.1
14.4
(a)
X is a product, because its concentration increases with time.
(b)
The average rate of reaction between any two points on the graph is the slope of
the line connecting the two points. The average rate is greater between points 1
and 2 than between points 2 and 3 because they are different stages in the overall
process. Points 1 and 2 are earlier in the reaction when more reactants are available, so the rate of formation of products is greater. As reactants are used up, the
rate of X production decreases, and the average rate between points 2 and 3 is
smaller.
Analyze. Given three mixtures and the order of reaction in each reactant, determine
which mixture will have the fastest initial rate.
Plan. Write the rate law. Count the number of reactant molecules in each container.
The three containers have equal volumes and total numbers of molecules. Use the molecule count as a measure of concentration of NO and O2. Calculate the initial rate for
each container and compare.
Solve. Rate = k[NO]2[O2]; rate is proportional to [NO]2[O2]
Container
[NO]
[O2]
[NO]2[O2] ∝ rate
(1)
5
4
100
(2)
7
2
98
(3)
3
6
54
The relative rates in containers (1) and (2) are very similar, with (1) having the slightly
faster initial rate.
14.6
Analyze. Given concentrations of reactants and products at two times, as represented in
the diagram, find t1/2 for this first-order reaction.
Plan. For a first order reaction, t1/2 = 0.693/k; t1/2 depends only on k. Use Equation
[14.12] to solve for k. Solve.
(a)
Since reactants and products are in the same container, use number of particles as
a measure of concentration. The red dots are reactant A, and the blue are product B. [A]0 = 8, [A]30 = 2, t = 30 min.
ln
= –kt. ln(2/8) = –k(30 min);
378
= k;
14 Chemical Kinetics
Solutions to Exercises
k = 0.046210 = 0.0462 min−1
t1/2 = 0.693/k = 0.693/0.046210 = 15 min
By examination, [A]0 = 8, [A]30 = 2. After 1 half-life, [A] = 4; after a second halflife, [A] = 2. Thirty minutes represents exactly 2 half-lives, so t1/2 = 15 min. [This
is more straightforward than the calculation, but a less general method.]
(b)
14.10
After 4 half-lives, [A]t = [A]0 × 1/2 × 1/2 × 1/2 × 1/2 = [A]0/16. In general, after
n half-lives, [A] = [A]0/2n.
This is the profile of a two-step mechanism, A → B and B → C. There is one intermediate, B. Because there are two energy maxima, there are two transition states. The B → C
step is faster, because its activation energy is smaller. The reaction is exothermic because the energy of the products is lower than the energy of the reactants.
Reaction Rates
14.13
14.15
(a)
Reaction rate is the change in the amount of products or reactants in a given
amount of time; it is the speed of a chemical reaction.
(b)
Rates depend on concentration of reactants, surface area of reactants, temperature and presence of catalyst.
(c)
The stoichiometry of the reaction (mole ratios of reactants and products) must be
known to relate rate of disappearance of reactants to rate of appearance of
products.
Analyze/Plan. Given mol A at a series of times in minutes, calculate mol B produced,
molarity of A at each time, change in M of A at each 10 min interval, and ΔM A/s. For
this reaction, mol B produced equals mol A consumed. M of A or [A] = mol A/0.100 L.
The average rate of disappearance of A for each 10 minute interval is
Solve.
Time (min)
Mol A
(a) Mol B
[A]
0
0.065
0.000
0.65
10
0.051
0.014
0.51
−0.14
2.3 × 10
20
0.042
0.023
0.42
−0.09
2
× 10
30
0.036
0.029
0.36
−0.06
1
× 10
40
0.031
0.034
0.31
−0.05
0.8 × 10
(c)
379
Δ[A]
(b) Rate −(Δ[A]/s)
–4
–4
–4
–4
14 Chemical Kinetics
14.17
(a)
Solutions to Exercises
Analyze/Plan. Follow the logic in Sample Exercises 14.1 and 14.2.
Time
(sec)
Time Interval
(sec)
Concentration
(M)
0
Solve.
ΔM
Rate
(M/s)
0.0165
–7
2,000
2,000
0.0110
–0.0055
28 × 10
5,000
3,000
0.00591
–0.0051
17 × 10
8,000
3,000
0.00314
–0.00277
9.23 × 10
12,000
4,000
0.00137
–0.00177
4.43 × 10
15,000
3,000
0.00074
–0.00063
2.1 × 10
(b)
–7
–7
–7
–7
From the slopes of
the lines in the figure at right, the
rates are:
at 5000 s,
12 × 10−7 M/s;
at 8000 s,
5.8 × 10−7 M/s
14.19
Analyze/Plan. Follow the logic in Sample Exercise 14.3.
(a)
–Δ[H2O2]/Δt = Δ[H2]/Δt = Δ[O2]/Δt
(b)
–Δ[N2O]/2Δt = Δ[N2]/2Δt = Δ[O2]/Δt
Solve.
–Δ[N2O]/Δt = Δ[N2]/Δt = 2Δ[O2]/Δt
(c)
–Δ[N2]/Δt = Δ[NH3]/2Δt; –Δ[H2]/3Δt = Δ[NH3]/2Δt
–2Δ[N2]/Δt = Δ[NH3]/Δt; –Δ[H2]/Δt = 3Δ[NH3]/2Δt
14.21
Analyze/Plan. Use Equation [14.4] to relate the rate of disappearance of reactants to the
rate of appearance of products. Use this relationship to calculate desired quantities.
Solve.
(a)
Δ[H2O]/2Δt = –Δ[H2]/2Δt = –Δ[O2]/Δt
H2 is burning, –Δ[H2]/Δt = 0.85 mol/s
O2 is consumed, –Δ[O2]/Δt = –Δ[H2]/2Δt = 0.85 mol/s/2 = 0.43 mol/s
380
14 Chemical Kinetics
Solutions to Exercises
H2O is produced, +Δ[H2O]/Δt = –Δ[H2]/Δt = 0.85 mol/s
(b)
The change in total pressure is the sum of the changes of each partial pressure.
NO and Cl2 are disappearing and NOCl is appearing.
–ΔPNO/Δt = –23 torr/min
+ΔPNOCl/Δt = –ΔPNO/Δt = +23 torr/min
ΔPT/Δt = –23 torr/min – 12 torr/min + 23 torr/min = –12 torr/min
Rate Laws
14.23
Analyze/Plan. Follow the logic in Sample Exercises 14.4 and 14.5. Solve.
(a)
If [A] is doubled, there will be no change in the rate or the rate constant. The
overall rate is unchanged because [A] does not appear in the rate law; the rate
constant changes only with a change in temperature.
(b)
The reaction is zero order in A, second order in B and second order overall.
(c)
14.25
Analyze/Plan. Follow the logic in Sample Exercise 14.4.
–3
Solve.
–1
(a)
rate = k[N2O5] = 4.82 × 10 s [N2O5]
(b)
rate = 4.82 × 10 s (0.0240 M) = 1.16 × 10 M/s
(c)
rate = 4.82 × 10 s (0.0480 M) = 2.31 × 10 M/s
–3
–1
–4
–3
–1
–4
When the concentration of N2O5 doubles, the rate of the reaction doubles.
14.27
Analyze/Plan. Write the rate law and rearrange to solve for k. Use the given data to calculate k, including units.
Solve.
(a, b) rate = k[CH3Br][OH−]; k =
(c)
14.29
–
–
Since the rate law is first order in [OH ], if [OH ] is tripled, the rate triples.
Analyze/Plan. Follow the logic in Sample Exercise 14.6.
–
Solve.
–
(a)
From the data given, when [OCl ] doubles, rate doubles. When [I ] doubles, rate
–
–
–
–
doubles. The reaction is first order in both [OCl ] and [I ]. rate = [OCl ][I ]
(b)
Using the first set of data:
381
14 Chemical Kinetics
Solutions to Exercises
(c)
14.31
Analyze/Plan. Follow the logic in Sample Exercise 14.6 to deduce the rate law. Rearrange
the rate law to solve for k and deduce units. Calculate a k value for each set of concentrations and then average the three values. Solve.
(a)
Doubling [NH3] while holding [BF3] constant doubles the rate (experiments 1
and 2). Doubling [BF3] while holding [NH3] constant doubles the rate (experiments 4 and 5).
Thus, the reaction is first order in both BF3 and NH3; rate = k[BF3][NH3].
(b)
The reaction is second order overall.
(c)
From experiment 1:
(Any of the five sets of initial concentrations and rates could be used to calculate
–1 –1
the rate constant k. The average of these 5 values is kavg = 3.408 = 3.41 M s )
(d)
14.33
–1 –1
rate = 3.41 M s (0.100 M)(0.500 M) = 0.1704 = 0.170 M /s
Analyze/Plan. Follow the logic in Sample Exercise 4.6 to deduce the rate law. Rearrange
the rate law to solve for k and deduce units. Calculate a k value for each set of concentrations and then average the three values.
Solve.
(a)
Increasing [NO] by a factor of 2.5 while holding [Br2] constant (experiments 1
2
and 2) increases the rate by a factor 6.25 or (2.5) . Increasing [Br2] by a factor of 2.5
while holding [NO] constant increases the rate by a factor of 2.5. The rate law for
2
the appearance of NOBr is: rate = Δ[NOBr]/Δt = k[NO] [Br2].
(b)
From experiment 1:
2
4
4
k2 = 150/(0.25) (0.20) = 1.20 × 10 = 1.2 × 10 M
2
4
4
–2
4
4
–2
k3 = 60/(0.10) (0.50) = 1.20 × 10 = 1.2 × 10 M
2
k4 = 735/(0.35) (0.50) = 1.2 × 10 = 1.2 × 10 M
4
–2
4
4
4
s
s
s
–1
–1
–1
4
kavg = (1.2 × 10 + 1.2 × 10 + 1.2 × 10 + 1.2 × 10 )/4 = 1.2 × 10 M
–2
s
–1
(c)
Use the reaction stoichiometry and Equation 14.4 to relate the designated rates.
Δ[NOBr]/2Δt = –Δ[Br2]/Δt; the rate of disappearance of Br2 is half the rate of appearance of NOBr.
(d)
Note that the data are given in terms of appearance of NOBr.
= 8.4 M/s
382
14 Chemical Kinetics
Solutions to Exercises
Change of Concentration with Time
14.35
14.37
(a)
[A]0 is the molar concentration of reactant A at time zero, the initial concentration
of A. [A]t is the molar concentration of reactant A at time t. t1/2 is the time required to reduce [A]0 by a factor of 2, the time when [A]t = [A]0/2. k is the rate
constant for a particular reaction. k is independent of reactant concentration but
varies with reaction temperature.
(b)
A graph of ln[A] vs time yields a straight line for a first-order reaction.
Analyze/Plan. The half-life of a first-order reaction depends only on the rate constant,
t1/2 = 0.693/k. Use this relationship to calculate k for a given t1/2, and, at a different temperature, t1/2 given k. Solve.
(a)
5
t1/2 = 2.3 × 10 s; t1/2 = 0.693/k, k = 0.693/t1/2
5
–6
k = 0.693/2.3 × 10 s = 3.0 × 10 s
(b)
14.39
–5
–1
–1
–5
–1
4
4
k = 2.2 × 10 s . t1/2 = 0.693/2.2 × 10 s = 3.15 × 10 = 3.2 × 10 s
Analyze/Plan. Follow the logic in Sample Exercise 14.7. In this reaction, pressure is a
measure of concentration. In (a) we are given k, [A]0, t and asked to find [A]t, using Equation [14.13], the integrated form of the first-order rate law. In (b), [At] = 0.1[A0],
find t. Solve.
(a)
lnPt = –kt + lnP0; P0 = 375 torr; t = 65 s
–2
–1
lnP65 = –4.5 × 10 s (65) + ln(375) = –2.925 + 5.927 = 3.002
P65 = 20.12 = 20 torr
(b)
Pt = 0.10 P0; ln(Pt/P0) = –kt
ln(0.10 P0/P0) = –kt, ln(0.10) = –kt; –ln(0.10)/k = t
–2
–1
t = –(–2.303)/4.5 × 10 s = 51.2 = 51 s
Check. From part (a), the pressure at 65 s is 20 torr, Pt ~ 0.05 P0. In part (b) we calculate
the time where Pt = 0.10 P0 to be 51 s. This time should be smaller than 65 s, and it is.
Data and results in the two parts are consistent.
14.41
Analyze/Plan. Given reaction order, various
values for t and Pt, find the rate constant
for the reaction at this temperature. For a
first-order reaction, a graph of lnP vs t is
linear with as slope of –k.
Solve.
t(s)
ln
0
1.000
0
2500
0.947
–0.0545
5000
0.895
–0.111
7500
0.848
–0.165
383
14 Chemical Kinetics
10000
0.803
Solutions to Exercises
–0.219
Graph ln
vs. time. (Pressure is a satisfactory unit for a gas, since the concentra–5 –1
tion in moles/liter is proportional to P.) The graph is linear with slope –2.19 × 10 s
–5 –1
as shown on the figure. The rate constant k = –slope = 2.19 × 10 s .
14.43
Analyze/Plan. Given: mol A, t. Change mol to M at various times. Make both first- and
second-order plots to see which is linear.
Solve.
(a)
time(min)
mol A
[A] (M)
ln[A]
1/mol A
0
0.065
0.65
–0.43
1.5
10
0.051
0.51
–0.67
2.0
20
0.042
0.42
–0.87
2.4
30
0.036
0.36
–1.02
2.8
40
0.031
0.31
–1.17
3.2
The plot of 1/[A] vs time is linear, so the reaction is second-order in [A].
(b)
For a second-order reaction, a plot of 1/[A] vs. t is linear with slope k.
k = slope = (3.2 – 2.0) M
–1
/ 30 min = 0.040 M
(The best fit to the line yields slope = 0.042 M
(c)
t1/2 = 1/k[A]0 = 1/(0.040 M
–1
–1
–1
min
–1
–1
min .)
–1
min )(0.65 M) = 38.46 = 38 min
(Using the “best-fit” slope, t1/2 = 37 min.)
14.45
Analyze/Plan. Follow the logic in Solution 14.43. Make both first and second order plots
to see which is linear. Solve.
(a)
time(s)
[NO2](M)
ln[NO2]
1/[NO2]
0.0
0.100
–2.303
10.0
5.0
0.017
–4.08
59
10.0
0.0090
–4.71
110
15.0
0.0062
–5.08
160
384
14 Chemical Kinetics
20.0
0.0047
Solutions to Exercises
–5.36
210
The plot of 1/[NO2] vs time is linear, so the reaction is second order in NO2.
(b)
–1
–1 –1
The slope of the line is (210 – 59) M / 15.0 s = 10.07 = 10 M s = k. (The slope
–1 –1
of the best-fit line is 10.02 = 10 M s .)
Temperature and Rate
14.47
14.49
(a)
The energy of the collision and the orientation of the molecules when they collide
determine whether a reaction will occur.
(b)
According to the kinetic-molecular theory (Chapter 10), the higher the temperature, the greater the speed and kinetic energy of the molecules. Therefore, at a
higher temperature, there are more total collisions and each collision is more
energetic.
Analyze/Plan. Given the temperature and energy, use Equation [14.18] to calculate the
fraction of Ar atoms that have at least this energy.
Solve.
f=e
–3.0070
= 4.9 × 10
–2
At 400 K, approximately 1 out of 20 molecules has this kinetic energy.
14.51
Analyze/Plan. Use the definitions of activation energy (Emax – Ereact) and ΔE (Eprod – Ereact)
to sketch the graph and calculate Ea for the reverse reaction. Solve.
(a)
(b)
385
Ea(reverse) = 73 kJ
14 Chemical Kinetics
14.53
14.55
Solutions to Exercises
Assuming all collision factors (A) to be the same, reaction rate depends only on Ea; it is
independent of ΔE. Based on the magnitude of Ea, reaction (b) is fastest and reaction
(c) is slowest.
Analyze/Plan. Given k1, at T1, calculate k2 at T2. Change T to Kelvins, then use the Equation [14.21] to calculate k2. Solve.
–2 –1
T1 = 20°C + 273 = 293 K; T2 = 60°C + 273 = 333 K; k1 = 2.75 × 10 s
(a)
(b)
14.57
Analyze/Plan. Follow the logic in Sample Exercise 14.11.
Solve.
3
k
ln k
T(K)
1/T(× 10 )
0.0521
–2.955
288
3.47
0.101
–2.293
298
3.36
0.184
–1.693
308
3.25
0.332
–1.103
318
3.14
3
The slope, –5.71 × 10 , equals –Ea/R. Thus,
3
Ea = 5.71 × 10 × 8.314 J/mol = 47.5 kJ/mol.
14.59
Analyze/Plan. Given Ea, find the ratio of rates for a reaction at two temperatures. Assuming initial concentrations are the same at the two temperatures, the ratio of rates will be
the ratio of rate constants, k1/k2. Use Equation [14.21] to calculate this ratio. Solve.
T1 = 50°C + 273 = 323 K; T2 = 0°C + 273 = 273 K
3
–4
ln (k1/k2) = 7.902 × 10 (5.670 × 10 ) = 4.481 = 4.5; k1/k2 = 88.3 = 9 × 10
1
The reaction will occur 90 times faster at 50°C, assuming equal initial concentrations.
Reaction Mechanisms
14.61
(a)
An elementary reaction is a process that occurs in a single event; the order is given
by the coefficients in the balanced equation for the reaction.
386
14 Chemical Kinetics
14.63
14.65
Solutions to Exercises
(b)
A unimolecular elementary reaction involves only one reactant molecule; the activated complex is derived from a single molecule. A bimolecular elementary reaction involves two reactant molecules in the activated complex and the overall
process.
(c)
A reaction mechanism is a series of elementary reactions that describe how an
overall reaction occurs and explain the experimentally determined rate law.
Analyze/Plan. Elementary reactions occur as a single step, so the molecularity is determined by the number of reactant molecules; the rate law reflects reactant stoichiometry.
Solve.
(a)
unimolecular, rate = k[Cl2]
(b)
bimolecular, rate = k[OCl ][H2O]
(c)
bimolecular, rate = k[NO][Cl2]
–
Analyze/Plan. Use the definitions of the terms ‘intermediate’ and ‘exothermic’, along
with the characteristics of reaction profiles, to answer the questions. Solve.
This is a three-step mechanism, A → B, B → C, and C → D.
14.67
(a)
There are 2 intermediates, B and C.
(b)
There are 3 energy maxima in the reaction profile, so there are 3 transition states.
(c)
Step C → D has the lowest activation energy, so it is fastest.
(d)
The energy of D is slightly greater than the energy of A, so the overall reaction is
endothermic.
(a)
(b)
Intermediates are produced and consumed during reaction. HI is the
intermediate.
(c)
Follow the logic in Sample Exercise 14.13.
First step: rate = k1[H2][ICl]
Second step: rate = k2[HI][ICl]
(d)
14.69
The slow step determines the rate law for the overall reaction. If the first step is
slow, the observed rate law is: rate = k[H2][HCl].
Analyze/Plan. Given a proposed mechanism and an observed rate law, determine which
step is rate determining. Solve.
(a)
If the first step is slow, the observed rate law is the rate law for this step.
rate = k[NO][Cl2]
(b)
Since the observed rate law is second-order in [NO], the second step must be
slow relative to the first step; the second step is rate determining.
387
14 Chemical Kinetics
Solutions to Exercises
Catalysis
14.71
14.73
14.75
(a)
A catalyst increases the rate of reaction by decreasing the activation energy, Ea, or
increasing the frequency factor A. Lowering the activation energy is more common and more dramatic.
(b)
A homogeneous catalyst is in the same phase as the reactants; a heterogeneous
catalyst is in a different phase and is usually a solid.
(a)
(b)
NO2(g) is a catalyst because it is consumed and then reproduced in the reaction
sequence. (NO(g) is an intermediate because it is produced and then consumed.)
(c)
Since NO2 is in the same state as the other reactants, this is homogeneous
catalysis.
(a)
Use of chemically stable supports such as alumina and silica makes it possible to
obtain very large surface areas per unit mass of the precious metal catalyst. This
is so because the metal can be deposited in a very thin, even monomolecular,
layer on the surface of the support.
(b)
The greater the surface area of the catalyst, the more reaction sites, and the greater the rate of the catalyzed reaction.
14.77
As illustrated in Figure 14.21, the two C–H bonds that exist on each carbon of the ethylene molecule before adsorption are retained in the process in which a D atom is added
to each C (assuming we use D2 rather than H2). To put two deuteriums on a single carbon, it is necessary that one of the already existing C–H bonds in ethylene be broken
while the molecule is adsorbed, so the H atom moves off as an adsorbed atom, and is
replaced by a D. This requires a larger activation energy than simply adsorbing C2H4
and adding one D atom to each carbon.
14.79
(a)
Living organisms operate efficiently in a very narrow temperature range; heating
to increase reaction rate is not an option. Therefore, the role of enzymes as homogeneous catalysts that speed up desirable reactions without heating and undesirable side-effects is crucial for biological systems.
(b)
catalase: 2H2O2 → 2H2O + O2; nitrogenase: N2 → 2NH3 (nitrogen fixation)
14.81
Analyze/Plan. Let k = the rate constant for the uncatalyzed reaction,
kc = the rate constant for the catalyzed reaction
According to Equation [14.20], ln k = –Ea /RT + ln A
Subtracting ln k from ln kc,
388
14 Chemical Kinetics
(a)
Solutions to Exercises
RT = 8.314 J/K-mol × 298 K × 1 kJ/1000 J = 2.478 kJ/mol; ln A is the same for
both reactions.
The catalyzed reaction is approximately 10,000,000 (ten million) times faster at
25°C.
(b)
RT = 8.314 J/K - mol × 398 K × 1 kJ/1000 J = 3.309 kJ/mol
The catalyzed reaction is 200,000 times faster at 125°C.
Additional Exercises
14.83
A balanced chemical equation shows the overall, net change of a chemical reaction.
Most reactions occur as a series of (elementary) steps. The rate law contains only those
reactants that form the transition state of the rate-determining step. If a reaction occurs
in a single step, the rate law can be written directly from the balanced equation for the
step.
14.86
(a)
2–
The rate increases by a factor of nine when [C2O4 ] triples (compare experiments
1 and 2). The rate doubles when [HgCl2] doubles (compare experiments 2 and 3).
2– 2
The rate law is apparently: rate = k[HgCl2][C2O4 ]
(b)
14.89
14.92
Using the data for Experiment 1,
–3
–2 –1
2
–5
(c)
rate = (8.672 × 10 M s )(0.100 M)(0.25 M) = 5.4 × 10 M/s
(a)
t1/2 = 0.693/k = 0.693/7.0 × 10 s = 990 = 9.9 × 10 s
(b)
k=
(a)
A = abc, Equation [14.5]. A = 0.605, a = 5.60 × 10 cm M , b = 1.00 cm
(b)
Calculate [c]t using Beer’s law. We calculated [c]0 in part (a). Use Equation [14.13]
to calculate k.
–4 –1
2
–4 –1
= 2.05 × 10 s
3
–4
–5
–1
–1
–4
–4
k = ln(1.080 × 10 ) – ln(4.464 × 10 ) / 1800 s = 4.910 × 10 = 4.91 × 10 s
(c)
For a first order reaction, t1/2 = 0.693/k.
389
–1
14 Chemical Kinetics
Solutions to Exercises
–4
–1
3
3
t1/2 = 0.693/4.910 × 10 s = 1.411 × 10 = 1.41 × 10 s = 23.5 min
(d)
At = 0.100; calculate ct using Beer’s law, then t from the first order integrated rate
equation.
3
3
t = 3.666 × 10 = 3.67 × 10 s = 61.1 min
14.95
ln k
1/T
–24.17
3.33 × 10
–20.72
3.13 × 10
–17.32
2.94 × 10
–15.24
2.82 × 10
–3
–3
–3
–3
4
The calculated slope is –1.751 × 10 .
The activation energy Ea, equals
– (slope) × (8.314 J/mol). Thus,
4
5
Ea = 1.8 × 10 (8.314) = 1.5 × 10 J/mol
2
= 1.5 × 10 kJ/mol. (The best-fit slope
4
4
is –1.76 × 10 = –1.8 × 10 and the val2
ue of Ea is 1.5 × 10 kJ/mol.)
14.99
(a)
Cl2(g)
2Cl(g)
(b)
Cl(g), CCl3(g)
(c)
Reaction 1 - unimolecular, Reaction 2 - bimolecular, Reaction 3 - bimolecular
(d)
Reaction 2, the slow step, is rate determining.
(e)
If Reaction 2 is rate determining, rate = k2[CHCl3][Cl]. Cl is an intermediate
formed in reaction 1, an equilibrium. By definition, the rates of the forward and
2
reverse processes are equal; k1 [Cl2] = k –1 [Cl] . Solving for [Cl] in terms of [Cl2],
Substituting into the overall rate law
(The overall order is 3/2.)
390
14 Chemical Kinetics
14.101
Solutions to Exercises
Enzyme: carbonic anhydrase; substrate: carbonic acid (H2CO3); turnover number:
7
1 × 10 molecules/s.
Integrative Exercises
14.104
–5
Analyze/Plan. 2N2O5 → 4NO2 + O2
–1
rate = k[N2O5] = 1.0 × 10 s [N2O5]
Use the integrated rate law for a first-order reaction, Equation [14.13], to calculate
k[N2O5] at 20.0 hr. Build a stoichiometry table to determine mol O2 produced in 20.0 hr.
Assuming that O2(g) is insoluble in chloroform, calculate the pressure of O2 in the 10.0 L
container. Solve.
ln[A]t – ln[A]0 = –kt; ln [N2O5]t = –kt + ln [N2O5]0
–5
–1
4
ln [N2O5]t = –1.0 × 10 s (7.20 × 10 s) + ln(0.600) = –0.720 – 0.511 = –1.231
[N2O5]t = e
–1.231
= 0.292 M
N2O5 was present initially as 1.00 L of 0.600 M solution.
mol N2O5 = M × L = 0.600 mol N2O5 initial, 0.292 mol N2O5 at 20.0 hr
2N2O5
t=0
→
4NO2
+
O2
0.600 mol
0
0
change
–0.308 mol
0.616 mol
0.154 mol
t = 20 hr
0.292 mol
0.616 mol
0.154 mol
[Note that the reaction stoichiometry is applied to the ‘change’ line.]
PV = nRT; P = nRT/V; V = 10.0 L, T = 45°C = 318 K, n = 0.154 mol
14.106
(a)
Use an apparatus such as the one pictured in Figure 10.3 (an open-end manometer), a clock, a ruler and a constant temperature bath. Since P = (n/V)RT, ΔP/Δt at
constant temperature is an acceptable measure of reaction rate.
Load the flask with HCl(aq) and read the height of the Hg in both arms of the
manometer. Quickly add Zn(s) to the flask and record time = 0 when the Zn(s)
contacts the acid. Record the height of the Hg in one arm of the manometer at
convenient time intervals such as 5 sec. (The decrease in the short arm will be the
same as the increase in the tall arm). Calculate the pressure of H2(g) at each time.
(b)
Keep the amount of Zn(s) constant and vary the concentration of HCl(aq) to de+
–
termine the reaction order for H and Cl . Keep the concentration of HCl(aq) constant and vary the amount of Zn(s) to determine the order for Zn(s). Combine
this information to write the rate law.
(c)
–Δ[H ]/2Δt = Δ[H2]/Δt; –Δ[H ]/Δt = 2Δ[H2]/Δt
+
+
391
14 Chemical Kinetics
Solutions to Exercises
[H2] = mol H2/L H2 = n/V; [H2] = P (in atm)/RT
+
Then, the rate of disappearance of H is twice the rate of appearance of H2(g).
14.109
(d)
By changing the temperature of the constant temperature bath, measure the rate
data at several (at least three) temperatures and calculate the rate constant k at
these temperatures. Plot ln k vs 1/T. The slope of the line is –Ea/R and Ea =
–slope (R).
(e)
Measure rate data at constant temperature, HCl concentration and mass of Zn(s),
varying only the form of the Zn(s). Compare the rate of reaction for metal strips
and granules.
In the lock and key model of enzyme action, the active site is the specific location in the
enzyme where reaction takes place. The precise geometry (size and shape) of the active
site both accommodates and activates the substrate (reactant). Proteins are large biopolymers, with the same structural flexibility as synthetic polymers (Chapter 12). The
three-dimensional shape of the protein in solution, including the geometry of the active
site, is determined by many intermolecular forces of varying strengths.
Changes in temperature change the kinetic energy of the various groups on the enzyme
and their tendency to form intermolecular associations or break free from them. Thus,
changing the temperature changes the overall shape of the protein and specifically the
shape of the active site. At the operating temperature of the enzyme, the competition
between kinetic energy driving groups apart and intermolecular attraction pulling them
together forms an active site that is optimum for a specific substrate. At temperatures
above the temperature of maximum activity, sufficient kinetic energy has been imparted so that the forces driving groups apart win the competition, and the threedimensional structure of the enzyme is destroyed. This is the process of denaturation.
The activity of the enzyme is destroyed because the active site has collapsed. The protein or enzyme is denatured, because it is no longer capable of its “natural” activity.
Cl• + Cl• → Cl2 is a termination step.
392